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The Lorentz invariant measure 

and the Heisenberg uncertainty principle 



BY PATRICK BRUSKIEWICH AND ALICIA KIEtBIK 

Abstract 

A novel approach to the Heisenberg uncer- 
tainty principle is presented in this paper in 
a fashion that directly links special relativity and 
quantum theory from first principles. 

A direct link between 
special relativity and 
quantum theory 

The traditional approach to non-relativistic quan- 
tum mechanics is primarily didactic. The basic 
axioms of the theory are introduced, along with 
the Heisenberg uncertainty principle and simple 
systems such as the quantum harmonic oscilla- 
tor, are elucidated. 

A different approach, which is heuristic in 
nature, is to begin with the Lorentz invariant mea- 
sure from special relativity. The Lorentz invariant 
measure can be used as a conceptual foundation 
of quantum mechanics, and a direct link between 
special relativity and quantum, mechanics can be 
presented without recourse to subsidiary con- 
ditions. As a result of this heuristic approach, a 
foundation can be laid that firmly joins the two 
fields, which are typically presented separately, 
until second quantization and the Klein-Gordon 
and Dirac equations are presented. 

The Lorentz invariant 
measure 

As outlined, in this paper, it appears evident thai 
special relativity and quantum mechanics can be 
directly linked from first principles. Starting with 
the Lorentz invariant measure, a novel approach to 
the Heisenberg uncertainty principle is presented. 
The Lorentz invariant measure is 1 



2 2 2 2 / 2 1 2 \ 

ex =c r -lx +y +z \, 



(1) 



which joins two points in Minkowski space. In 
this paper, we can look at the xt-plane without 
any loss of generality. For a one-dimensional 
inertial system, 

2 2 2^2 2 f-)\ 

C T =C t~ -x . U) 

Multiplying through by mass m, and rear- 
ranging terms, we get 

2 2.2 22 / n \ 

mx -mc i = -mc x . (3) 



Take the partial derivative of this expression 
with respect to X, the proper time ("temps proper" 
which literally means one's own time) and sim- 
pl i fy. This yields 



dx 2 dt 

xm tmc — = —mc " x. 

dx dx 



14) 



From special relativity, m(dx/dx) = my(3x/9t) = 
ymv = p, and (dt/dx) = y, so then 

xp ~Et = -mc~% (5) 

is also invariant. In 3D, this is the familiar inner 
product of the two four- vectors, namely 

x Pm p =-mc 1 x. (6) 

In quantum mechanics, this measure is impor- 
tant when describing a plane wave (unction 



\|/ =exp 



i(px—Et) 

ft 



= exp 



i*-p>) 



(7! 



or a wave packet made up of a superposition of 
states such as 



j/K/?)exp 



i^p») 



d N p, (8) 



w T here A(p) is a momentum dependent amplitude, 
and N is the dimension of the space. 
By inspection, it is evident, then, that 



x P, 



(9) 



represents a phase (p. 

Measurement in quantum mechanics involves 
both amplitude and phase considerations. For 
information to be pass between two events in 
space-time a signal must be exchanged between 
the two events. The fastest signal that can be 
exchanged travels at the speed of light. As a signal, 
or wave, travels between two points in space, its 
change in phase may be used as a measure of the 
separation between the two points in space-time. 

Phase difference 
between adjacent events 
in Minkowski space-time 

In studying quantum measures, we can use dif- 
ferent approaches, such as the limiting process 
and infinitesimal analysis, a technique developed 
at Cambridge beginning in Newton's time. In 
this paper, we shall use. infinitesimal analysis. 2 



In recent times, some of the infinitesimal analy- 
sis techniques have also been used in Wiener 
measures and Ito calculus in stochastic quantum 
mechanics. 

In ID, the difference in phase between two 
adjacent events in Minkowski space-time, two 
points infinitesimally separated by a causal 
connection (i.e. two points on or within the light 
cone connecting the two events) is 3 

5(p « <f> 2 -<p, = (x 2 p 2 - E 2 t 2 ) - (x, p } - E,t\ ) 
= (x l +§x)(p l +dp)-{E l +bE)(t l +8t) 

= bxp { +xfip+ 8x8p - 3Et } - E { 8t - 8E8t. 

(10) 
This expression can be grouped into three 
terms First, consi der the bEt } term. Given thai 
E - • N /pV+ m V , then 



5£ - 1 5p = v 5p - _J- qV, 

E t 



(ID 



where we have used p = ymv and E = ymc 2 . Since 

v = 5x/5t, this means vt, = x } . So for two points 

infinitesimally separated by a causal connection, 

X[8p - 5Et[ - and these two terms cancel each 

other out. in the expression. 

Consider the E,or term: 

( -< 2 ^ 
£,5/ = ymc 2 3t = (ymv ) — 6/ = pfix, 

\ i J 

(12) 
where we have used the group velocity v g given 
by v = c 2 /v g . For a wave disturbance, whether it is 
a massless particle like a photon or a massive par- 
ticle like an electron, it is the group velocity that 
characterizes the momentum or energy o{ the 
w T ave packet, and therefore the exchange of infor- 
mation between two points in space-time. For 
two points infinitesimally separated by a causal 
connection, Efit - p { 3x = and these two terms 
cancel each other out in the expression. 
Of the six terms in the infinitesimal phase 
expression, only two terms remain: the differ- 
ence in phase between two points infinitesimally 
separated by a causal connection in space-time 
given by 

8(p~5x6>-8£5/. (13) 

From the standpoint of infinitesimals, this 
expression is counter-intuitive in that the phase 
is made up of a product of terms and we would 
expect an infinitesimal change in the phase to be 
first order in these terms. The fact the first-order 



16 Canadian Undergraduate Physics Journal 



Volume VI Issue 3 APRlt2008 



infinitesimal terms drop out of the expression 
is a direct result of the Lorentz invariant mea- 
sure and appears to form the conceptual basis 
to the Heisenberg uncertainty principle. It is at 
this juncture that special relativity and quantum 
mechanics are linked at first principles. 

For a massless particle like a photon, then 

8E - cSp => 8Ebt = Spcdt = dpdx. (14) 

As is evident, near an extremum where Sep is 
small, then 



dEdt « 8p8x. 



(15) 



Based on our derivation from the invariant 
measure, there is nothing that predisposes that 
Sep = 0, 8E8t ~ 0, or 8p8x ~ 0. When we consider 
that information exchange requires the exchange 
of energy or momentum, then by dimensional 
analysis, we find 

8E8t « 5>Sx ~ h. (16) 



The classical standpoint 

Let us consider this expression from a classical 
standpoint, in terms of generalized momentum 
p and generalized q. By the equivalence of work- 
energy, we have 



\dE=\Fdq=\^d q . 



(17) 



We consider the phase space of a quasi-sta- 
tionary or periodic system and take a time aver- 
age 



{dp . 



J**-* J**- 



(18) 



cance of this remarkable constant." 1 

The Dfrac delta potential 

It is straightforward to use this quantum action to 
solve for the energy states of a particle in a box or 
for a quantum harmonic oscillator. A more chal- 
lenging system to studying using the phase space 
approach is that of a particle trapped by a Dirac 
delta potential. 

Consider a potential of finite size and shrink 
the region of interaction to a point. Expressed as 
a ID attractive Dirac delta potential of strength a 
the Schrodinger equation for this system is 

ft 2 d 2 
- \|/-a5(x)y = £\|/. (21) 

2m ax 

There are a number of ways to solve this second 
order di fferential equation, including using integral 
transforms. The integral transform methods are 
outlined in a paper titled "A Novel Look at the One 
Dimensional Delta Schrodinger Equation/' written 
by one of the authors for the Gamma Magazine of 
the Niels Bohr Institute in May 2005. 5 

As is well known, this equation has the single 
solution 



i|/(x) = \|/(0)exp 

By inspection, 
px ma\x\ 



-ma x 



ma 



(22) 



(23) 



ft ft 2 n 

and so we can solve for the energy of this state, 
namely 



N- 



di J 5/ 

This implies then that 

5/ jdE » jdpdq ^> StdE w dpdq. (19) 

The classical concept of phase space is consis- 
tent with the derivative from the Lorentz invari- 
ant measure. 

At the Solvay Conference in 1911, Max Planck 
introduced the quantum action, namely 

jdpdq = nth (20) 

where n is an integer and ti is Planck's constant. 
This phase integral represents the quantization of 
quantum action. Planck expressed the view that 
"one should therefore confine oneself to the prin- 
ciple that the elementary region of probability, ft 8p8x 
has an ascertainable finite value and avoid any 
further speculation about the physical signifi- 

WWW.CUPJ.CA 



{ma I h) 



tmx^ (24) 

2m 2m 2ft 2 

The velocity of the particle can be derived 



(25) 



from 

ma a 8x 

p = mv = => v - — = — . 

ti ft 8t 

'fake the derivative of the momentum with 
respect to the potential strength a: 

8am 



8p = - 



(26) 



if we multiply both sides of this equation with 
ox = abl/ft, then 



_( 8am Va5/ 



"I » A » 



(27) 



\ 



We see that while we are dealing with a 
delta potential, what this last expression implies 
is that neither 5p6x — » nor 8E8t —> for this 
system. Infinitesimal analysis of this system 
shows that 

5£5/ a 6>5x « ft, (28) 

as expected. 

Conclusion 

The Lorentz invariant measure can be used as a 
conceptual foundation of quantum mechanics, 
and a direct link between special relativity and 
quantum mechanics can be presented without 
recourse to subsidiary conditions. In this paper, 
using infinitesimal analysis, a technique devel- 
oped at Cambridge beginning in Newton's time, 
we have derived the Heisenberg uncertainty 
principle from first principles. 

Acknowledgements 

Patrick would like to thank Dr. J. McKenna and 
Dr. M. McMillan for their valuable insights on 
the Lorentz invariant measure, and both Dr. F. 
Kaempffer and Dr. A. Zhitnisky for their unique 
insights on the Heisenberg uncertainty prin- 
ciple. 

References 

] C. Moller. The Theory of Relativity, Oxford Uni- 
versity Press. 1972, p. 97. 

2 J.L. Bell. A Primer of Infinitesimal Analysis. Cam- 
bridge University Press: Cambridge. 1998, 
p. 29. 

3 R. Schlegel. Superposition and Interaction. Univer- 
sity of Chicago Press: Chicago. 1980, p. 198. 

l M. Jammer. The Conceptual Development of 
Quantum Mechanics. McGraw Hill: New York. 
1966, p. 54. 

5 P. Bruskiewich. "A Novel Look at the One 
Dimensional Delta Schrodinger Equation." 
Gamma Magazine. 138 (May 2005). 



Alicja Kielbik is a third year undergrad at the 
University of British Columbia (UBC). She is 
interested in quantum continuous measurement 
theory and its application to quantum computing 
and gravity. 

Patrick Bruskiewich is a UBC doctoral can- 
didate and editor-in-chief of CUPj. He can be 
reached at patrichh@phas.uhc.ca. 



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Canadian Undergraduate Physics Journal 1 7